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let $f$ and $f_{n}$ are non-negative extended real-valued measurable functions and $f_{n}(\omega)\nearrow f(\omega)$, $\forall \omega \in \Omega $ ,then $$\int f\quad d\mu=\sup_{n} \int f_{n}\quad d\mu=\lim_{n\to \infty}\int f_{n}\quad d\mu$$ thanks for help.

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The second equality follows from the fact that the limit of a nondecreasing sequence is the supremum of the sequence along with the fact that integration preserves inequalities.

The first equality follows from the monotone convergence theorem, a standard result of measure theory.

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